Design and operation of a flat linear induction pump

7-1964

Design and operation of a flat linear induction

pump

Delwyn Donald Bluhm

R. W. Fisher

J. W. Nilsson

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Recommended Citation

Bluhm, Delwyn Donald; Fisher, R. W.; and Nilsson, J. W., "Design and operation of a flat linear induction pump" (1964).Ames Laboratory Technical Reports. 92.

Abstract

A five pole, one slot per phase, full pitch, twelve coil, wye connected, three phase linear induction pump has been designed for operation with 400 C sodium. The design incorporated a graded winding in order to reduce the net pulsating component of the magnetic flux density wave and a pump duct using copper side bars. The small laboratory-size pump was designed in two steps: stator design and winding design. First, the stator and core dimensions were established on the basis of maximum pressure development and maximum power at the design flow and slip. Maximum pressure output was a requirement of the intended pump application. The condition of maximum power output was used in order to provide a good pump efficiency. Second, the parameters for the induction pump equivalent circuit were determined for several winding methods. The winding method which offered 3920 ampere turns per phase (equivalent to a flow rate of 4. 88 gpm, a pressure of 60 psi, a line voltage of 250 volts, and a wave length of 21.2 cm as calculated in the first step) was selected. The theoretical performance characteristics of the flat induction pump were computed using the equivalent circuit parameters for a winding with 60 turns per coil. The measurement of developed pressure (0 to 42. 13 psi), sodium flow rate (0 to 9. 93 gpm), and power input (0 to 103. 9 watts) after installation of the pump in an operational sodium circulating loop provided an experimental determination of the pump performance characteristic. In addition, the effect of pump duct temperature on sodium flow rate and a differential transformer system used to locate the sodium level in manometer columns were briefly discussed .

Disciplines Engineering

IOWA STATE UNIVERSITY

DESIGN AND OPERATION OF A FLAT LINEAR INDUCTION PUMP

by

Delwyn Donald Bluhm, R. W. Fisher and J. W. Nilsson

PHYSICAL SCIENCES REflJ>ING R

~

&~~~ ~£[ID®~&1J®~W

RESEARCH AND

DEVELOPMENT

REPORT

Engineering and Equipment (UC-38} TID-4500, September 1, 1964

UNITED STATES ATOMIC ENERGY COMMISSION Research and Development Report

DESIGN AND OPERATION OF A FLAT LINEAR INDUCTION PUMP

by

Delwyn Donald Bluhm, R. W. Fisher and J. W. Nilsson

July, 1964

Ames Laboratory at

Iowa State University of Science and Technology F. H. Spedding, Director

IS-991

This report is distributed according to the category Engineering and Equipment (UC-38), as listed in TID-4500, September 1, 1964.

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TABLE OF CONTENTS

LIST OF SYMBOLS

Optimum Stator Structure

Winding Design

ABSTRACT

I. INTRODUCTION

II. ELECTROMAGNETIC PUMPS

III. REVIEW OF DESIGN THEORY

A. Optimum Stator Structure

B. Windings

IV. THEORETICAL DESIGN OF PUMP

A. Optimum Stator Structure

B. Windings

C. Theoretical Performance Characteristics

V. INVESTIGATIONS

A. Equipment

B. Experimental Procedure

VI. EXPERIMENTAL RESULTS

VII. CONCLUSIONS AND SUMMARY

VIII. LITERATURE CITED

IX. APPENDIX

A B ~ d f F Fa Fm g H I

Im, im

LIST OF SYMBOLS

Optimum Stator Structure

rms ampere conductors per unit length

instantaneous gap flux density

amplitude of gap flux density

equivalent pump duct diameter

frequency

instantaneous value of armature mmf wave

amplitude of armature mmf wave

magnetizing component of F

total magnetic gap

head loss

rms stator phase current

amplitude and instantaneous yalues of magnetizing component

of Is per unit length

amplitude and instantaneous values of liquid current wave,

power component of Is per unit length

amplitude and instantaneous values of total equivalent stator

surface current density per unit length

amplitude and instantaneous values of duct current wave,

power component of Is per unit length

kinematic viscosity of fluid

winding distribution factor

winding factor, kd kp

K pump parameter,

pump parameter to give maximum value of zh and z0 at constant

s and X

pump parameter for optimum design, s0

=

=

X constant

1 distance measured in direction of induced currents

n number of poles

NPH total series turns per phase

p average pressure developed per unit length of pump

(p)actual actual average pressure developed, (xp - P£)

P£ hydraulic friction loss (experimentally determined)

P power loss in the liquid metal per unit length

Pi input power to liquid metal

P₀ ideal power output in the liquid metal per unit length

(P0)actual actual power output in the liquid metal (total)

power loss in the pump duct wall per unit length

q number of phases

Q total flow

Reynolds number

resistance of liquid parallel to Ir

combined resistance of top and bottom pump duct walls

slip for maximum efficiency at constant X

slip for maximum pressure at constant Is' X and K

slip for maximum developed power at constant X and K

t internal thickness of pump duct parallel to magnetic field

thickness of pump duct wall

v velocity of liquid metal in pump duct

relative velocity between liquid and synchronous field

w

X

X

velocity of magnetic field

internal width of pump duct parallel to current flow

distance measured in direction of flow

P

wall loss parameter, Pw

t

head utilization factor, non-dimensional part of formula for p

maximum value of zh with variable K

maximum value of zh with variable s

output utilization factor, non-dimensional part of formula

z₀k maximum value of z₀ with variable K

z08 maximum value of z0 with variable s

a,~ phase angles

(~)actual actual operating efficiency

~e electrical efficiency

~em maximum electrical efficiency

~h hydraulic efficiency

8

p

CT

cos

8

E~

f

g

I

internal phase angle between Is and Ir + Iw

viscosity of liquid metal

wavelength or double pole pitch

resistivity of liquid

resistivity of pump duct wall material

density of liquid metal

internal power factor

Winding Design

area of gap orthogonal to magnetic field (i.e., L wB) square

inches

cross sectional area of conductor, square inches

cross sectional area of copper side bars, square inches

circuits in parallel

depth of slot occupied by conductors, inches

conductor free depth of slot, inches

equivalent diameter of copper side bars (i.e.

-1-)

.,.

line to line voltage, volts

phase to neutral voltage, volts

frequency, cps

total magnetic gap, inches

rms stator phase current, amperes

magnetizing current, amperes

winding distribution factor

L

(MLC)

n

s

t

t w

w

length of magnetic field, pump or pump duct over wound region,

inches

mean length of conductor, inches

number of poles

primary conductors in series per phase

number of phases

primary resistance per phase, ohms

equivalent resistance of fluid per phase referred to stator,

ohms

equivalent resistance of duct walls per phase referred to

stator, ohms

slip,

number of stator slots containing copper

total slots in iron

internal thickness of pump duct parallel to magnetic field,

inches

thickness of pump duct wall, inches

internal width of pump duct parallel to current flow, inches

effective field width (i.e. iron width x stacking factor),

·inches

slot width, inches

primary leakage reactance, ohms at f

magnetizing reactance, ohms at f

p

~u

Pw

resistivity of liquid, microhm inches

resistivity of copper, microhm inches

DESIGN AND OPERATION OF A FLAT LINEAR INDUCTION PUMP···

Delwyn D. Bluhm, R. W. Fisher and J. W. Nilsson

ABSTRACT

A five pole, one slot per phase, full pitch, twelve coil, wye

connected, three phase linear induction pump has been designed for

operation with 400 C sodium. The design incorporated a graded

wind-ing in order to reduce the net pulsating component of the magnetic flux

density wave and a pump duct using copper side bars. The small

laboratory-size pump was designed in two steps: stator design and

winding design. First, the stator and core dimensions were

establish-ed on the basis of maximum pressure development and maximum power

at the design flow and slip. Maximum pressure output was a

require-ment of the intended pump application. The condition of maximum

power output was used in order to provide a good pump efficiency.

Second, the parameters for the induction pump equivalent circuit were

determined for several winding methods. The winding method which

offered 3920 ampere turns per phase (equivalent to a flow rate of 4. 88

gpm, a pressure of 60 psi, a line voltage of 250 volts, and a wave

length of 21.2 em as calculated in the first step) was selected. The

theoretical performance characteristics of the flat induction pump

were computed using the equivalent circuit parameters for a winding

with 60 turns per coil. The measurement of developed pressure (0 to

42. 13 psi), sodium flow rate (0 to 9. 93 gpm), and power input (0 to 103. 9

watts) after installation of the pump in an operational sodium circulating

loop provided an experimental determination of the pump performance

characteristic. In addition, the effect of pump duct temperature on

sodium flow rate and a differential transformer system used to locate

the sodium level in manometer columns were briefly discussed .

.,_

.,.

I. INTRODUCTION

Devices which rely on the force that is produced whenever a current

carrying solid conductor is imbedded in a magnetic field are well known

to today's engineers and scientists. Not so well known are the two

groups of devices in which the current is carried either in a conductive

gas or a conductive liquid.

An example of a device using a conducting gas is the MHD

(Magneto-hydrodynamic) generator (1). The gas is made conductive by partial

ionization and is forced to cut a magnetic field. Electrodes

perpendic-ular to the field draw off the induced currents. Electromagnetic pumps

and flowmeters are two very good examples of devices.using conductive

liquids. The conductive liquids are usually molten metals at high

temperatures. The electromagnetic pump provides the current and ~gnetic

field in the conductive liquid. The resulting force which causes the

liquid to be pumped is the useful output. The electromagnetic flowmeter

operates on the same principal as the MHD generator except

that the conductive mediums are different. The induced EMF at the

electrodes is proportional to the rate of flow of the conductive liquid.

This work deals primarily with electromagnetic pumps. These pumps

have become increasingly important in the last few years due to the

expanded use of molten metals as heat-transfer mediums for nuclear power

plants. Pumps developed for this purpose are normally very large. For

example,an A. C. linear induction pump for Argonne's Experimental Breeder

Reactor II (2) has a capacity of 5,000 gpm of 700 F sodium at 40 psi

less frequently used to circulate liquid metals and alloys in corrosion test loops (3) and to feed aluminum to a die-casting machine (4).

The object of this work was to design and determine the performance

II. ELECTROMAGNETIC PUMPS

In 1907 Northrup (5) developed a hydrostatic pressure of several

inches of water via the interaction of a 600 ampere current and a

magnetic field in mercury. He postulated that this effect could be

utilized in an ammeter for measuring large currents.

The movement of a liquid metal by using a sliding and rotating

magnetic field was proposed by Chubb (6) in 1915. Bainbridge (7) tested

an alternating current conduction pump in 1926. Einstein and Szilard

proposed an annular linear induction pump for use with alkali metals in

1928. Several working models of this pump rated at about 2 kw were

operated with mercury and liquid alloys of sodium and potassium. The

design equations for this type of pump using liquid bismuth were

pre-sented in a report by Feld and Szilard (8) in 1942.

The flat linear induction pump is only one of many types of

electro-magnetic pumps for liquid metals. The principal of operation is the same

for all electromagnetic pumps. It is the source of the current and

magnetic field that changes. Electromagnetic pumps can be divided in

two main categories; they are the conduction pumps and the induction

pumps.

Conduction pumps may be operated either on alternating or direct

current. Either type pump may be identified by the electrodes, brazed

to the pump duct wall, which supply the current from an external source

to the liquid metal. Conduction pumps have one primary advantage over

the other electromagnetic pumps; they can be used to pump even the

high density, and high viscosity. The current flowing through the liquid

metal may be made as large as is necessary to overcome the effects of the

heavy metals. The biggest disadvantage of these pumps is the requirement

of extremely large currents, usually several thousand amperes at volta~es

of one or two volts.

A direct current conduction pump can be operated either with a

permanent magnet or an electromagnet to produce the magnetic field. The

disadvantage of the awkward excitation requirements, as mentioned earlier,

is partially overcome by the use of a relatively efficient homopolar

generator with liquid metal brushes. The use of conventional rectifiers

is not recommended for the larger pumps since they are inefficient,

bulky, and expensive. Another requirement (and not always a disadvantage)

for satisfactory operation of a direct current pump is the wetting of

the duct wall by the liquid metal. The performance characteristics of

the direct current conduction pump are excellent as the pump is applicable

in systems requiTing wide variations in power, pressure, and flow. In

efficiency too, the direct current pump is unsurpassed by the other types

of electromagnetic pumps.

In order to avoid the use of a homopolar generator, an alternating

current conduction pump with its associated step-down transformer is

sometimes used. The immediate problems encountered with this pump are:

lamination of magnetic circuit to reduce eddy-currents and iron losses

and thereby improve the normally low efficiency; physically laTger pump;

poor power factor; and noisy operation. The alternating current

with size at a constant frequency. However, larger pumps may be built

if the frequency is sufficiently lowered. Normally, the reliable

step-down transformer is employed near or is part of ~he pump,while alternate

sources of supply may be located remotely.

The second category of electromagnetic pumps, the induction pumps,

always have alternating current induced in the molten metal by a

travel-ing or pulsattravel-ing magnetic field. These pumps are normally, with one

exception, used only with the typical light metals with higher

conduc-tivities so that the induced currents will be sufficiently large. A

classification of the most common types is:

Single-phase annular

Polyphase helical

Polyphase flat linear

Polyphase annular linear

Rotating magnet

The single-phase annular induction pump is similar to a transformer

with stationary primary winding but a moveable secondary winding. The

liquid metal acts as the secondary winding, and it is pumped by the

interaction of radial primary magnetic field in the annular gap and

the circulating induced current in the liquid. This pump is generally

used only for small pumping requirements because of its inherent low

efficiency and poor power factor. For larger pumping capacities the

The polyphase helical induction pump employs a rotating magnetic

field produced in the same manner as a squirrel cage induction motor.

The squirrel cage is replaced by a cylindrical liquid metal duct in an

increased air gap surrounding a "blocked" cylindrical iron core. Helical

guide vanes are placed in the duct to impart axial motion to the liquid

which is rotating from the action of the rotating magnetic field produced

by the three-phase stator winding. The function of end rings is now

served partially by the liquid and partially by two copper rings,silver

soldered to the duct at each end of the core. A helical induction pump

of this type normally is used in applications requiring high pressure

and low flow.

The most popular polyphase induction pump is the flat linear type.

The resulting linear fluid motion is an improvement upon the helical

induc-tion pump when large flows are required. The flat linear inducinduc-tion pump

is most easily described as a helical pump which has been cut open axially

along one side and finally straightened into a flat rectangular shape.

The duct is now located between two axially laminated structures. To

' improve pump performance,copper side bars are usually added to each side

of the duct; they function as end rings in a conventional induction motor.

The traveling field is produced by the polyphase multipole stator

wind-ings located on either one or both of the laminated iron structures. The

losses in this pump may be substantially reduced by grading the magnetic

field in the end sections. Grading allows the magnetic field to gradually

increase from near zero at the ends to the uniform value over the fully

An important feature of this pump is the possible removal of the stator

structures and windings without disturbing the liquid metal system.

The annular linear pump (or Einstein-Szilard pump) can be thought of

as being formed from the flat linear type by bending the latter into a

cylindrical shape. The radially laminated, cylindrical core, now on the

inside of an annular duct, has no windings. The windings,which are

pan-cake-like coils,are located in the circular slots of the stator

surround-ing the pump duct. Copper side bars are not needed since the induced

currents in the liquid metal flow in circular closed-loop paths. The

reduction of end effects by field grading is essential for good

perform-ance. As compared to the flat linear pump, the annular pump offers lower

hydraulic loss, stronger and less complicated construction, lower heat

loss, easier field grading, higher leakage reactance, and a more

compli-cated winding removal procedure.

Rotating field induction pumps induce an alternating current in the

liquid metal through the use of mechanically rotated permanent magnets

or electromagnets. All of the polyphase induction pumps with rotating

magnetic fields can be modified to form a rotating magnet pump. The

usual arrangement is to use rotating magnets for pumps with loop or helix

pump ducts. Copper side bars are required. This type of induction pump

has several-advantages over the other types, such as a stronger available

field providing the possibility of pumping high-resistivity metals, the

near unity power factor operation of induction motors, self-cooled magnets,

good efficiency and variable pole pitch. The drawbacks include the use of

The linear induction type of electromagnetic pump was selected since

it has proven its ability to operate continuously at temperatures near

1000 C for as long as 5000 hours (9). This is a very important capability

when a pump is to be used in corrosion test loops for circulating metals

with high melting points. This selection was also made since it was hoped

that this work would justify the use of small flat linear induction pumps

in laboratory applications where the pumping requirements are normally

very small. In the literature the flat linear induction pump was most

commonly employed at large flows for high efficiency while the single

phase induction pump and the rotating magnet induction pump were the two

types most frequently designed for laboratory use. The efficiencies of

these two types of pumps are generally low at the low flow rates where

they are employed. The use of the flat linear induction pump offers the

advantage of a simpler fabrication technique than with the single phase

induction pump; however, the efficiency is slightly less. It offers

several advantages over the rotating magnet pump,such as the elimination

of moving parts and a simplified pump duct removal procedure. Even

though the efficiency of a flat linear induction pump decreases at low

flow, proper design of the pump should yield an acceptable .efficiency.

In fact,this efficiency should compare favorably with the efficiencies

of the other small pumps under similar operating conditions.

Normally, the efficiency of linear induction pumps for corrosion

loops is unimportant. However, efficiency is a consideration in test

loops which are to be operated under design conditions of flow rate and

with an efficiency of 0.12% (9) or lower could cause the heat input to

the molten metal to be excessive. This fact necessitated the design of

a more efficient flat linear induction pump at a preselected flow rate.

The pump was designed specifically for a working fluid of sodium. Sodium

is one of the best metals for use with linear induction pumps since it is

a typical light metal with low resistivity, low density, and low

viscosity.

The availability of an operational sodium loop provided a means of

III. REVIEW OF DESIGN THEORY

Flat linear induction pumps are sometimes thought to be analogous to

squirrel cage induction motors. This is true when considering only the

basic principles of the operation of each; however, the similarity ends

there. Linear motion rather than rotary motion is required from these

pumps, and this causes many different winding design problems. There are

many compromises between electrical, physical, and hydraulic requirements

which increase the losses and decrease the power factor. For example:

high fluid velocity yields a minimum pump size with maximum power factor

and larger hydraulic losses; tooth saturation limits the magnetic field

strength; deep slots required for cooling of coils reduce the power factor

and air gap flux density; and heat insulation surrounding the pump duct

increases the air gap and further reduces the power factor and flux

density.

The pump design theory is presented in two parts. The first part

deals with the development of equations for currents in the duct and

Liquid, pressure developed, power output, and efficiency after particular

values of stator ampere turns or magnetic field strength have been

assumed to be available. Some of these equations may be maximized with

respect to slip or to a factor called the pump parameter which depends on

the physical dimensions of the pump. The stator structure is called an

optimum structure for a given set of pumping requirements when it has

been optimized with respect to slip or pump parameter for maximum pressure

developed, power output or efficiency,depending on what capabilities

selecting a winding configuration which will produce the ampere turns or

magnetic field strength assumed in the first part.

The design of a flat linear induction pump is quite involved due to

the large number of variables. The variables of major concern are liquid

metal, pump duct material, temperature of operation, wavelength, duct

thickness and width, total magnetic gap, electrical frequency, total

number of poles, flow rate, pressure developed, power output in fluid,

and overall efficiency. Many of these variables are assumed or selected

in order that the other parameters may be calculated from equations

derived for maximum pressure, power output or efficiency. If these

results are physically realizable, the assumed, selected, and calculated

values are used in the design.

It is extremely difficult to theoretically account for the many

losses in linear induction pumps. Equations derived for the power losses

in the liquid metal, the duct wall, and the stator winding give a

reason-able indication of pump efficiency since other losses are normally small

in comparison. The "rotor" reactance in the following sections is

assumed negligible. It is further assumed that the net pulsating

compo-nent of the magnetic flux is zero; this is a good assumption if the

winding is properly graded (i.e. partially wound at the pump ends causing

the field to gradually increase from zero).

A. Optimum Stator Structure

The following is a summary of flat linear induction pump design

in the field of electromagnetic pumping and has written many reports covering the design of various types of electromagnetic pumps.

The traveling wave equation of the equivalent stator surface current density or current sheet can be written in the form

is = Is sin 2 Tr ( ~ ft) (1)

The mutual magnetic flux in the liquid metal is derived from a magnetiz-ing current wave

im = Im sin 2 Tr (-L - ft

-a )

).

Since the currents in the liquid and duct walls are in phase, they are

ir + iw

=

+

X -

+

{3)

The rotor and magnetizing currents are different in phase by

a

+/3

Therefore

= J!_

2.

The amplitude of the mmf wave for a three phase distributed frac-tional pitch winding is

Since

F

=

2

=

I= 2.70

kw

amp-turns/phase

pole

=

A 2

The traveling wave equation of the mmf wave is

F = Fa sin 2 Tr (

~

~

The total stator surface current density wave as a function of Fa can be

obtained by differentiating Equation 9 with respect to x.

is =

..2!!..

_A

~

Comparing Equation 10 with Equation

=~F

~JZ

Is

_A

_A

or using equation (6)

If q

=

I _s

=

1 yields

~

A

A

2 2q

I

=

.fi

The result is

(10)

kwA

The correspondence between Fm and Im (Equation 2) in the fully wound

region is similar to that which exists between F (Equation 9} and is

(Equation 10). Therefore

The flux density is

F

=

m

p.o

=

_A_

B _m

=

B g cos 2., (~ - ft

-a

m

A

X

Im cos 27r

(T-

-a

P.o

A

2 ., g (15)

The components of I are very important as they show the division of

useable and unuseable current between the liquid metal and pump duct wall.

It is clear that

Bm lvr Bm w(v8 - v) Bm SV 8 t

Ir

=

=

Rr p w/t p

Since

Equations 15 and 16 may now be combined to

Im 271' g

p

p

=

P.o

~

-Ir sv t sft sK

(17)

where the pump parameter, K, is defined

P.o

A

K =

z.,.

p

Likewise

But

I _w = = =

Rw

(19)

From Equations 16 and 19

Iw

p

- - =

=

-Ir s Pw t s (20)

where the wall loss parameter, X, is defined

X= ₍₂₁₎

Excluding stator leakage reactance, the internal power factor from

Equa-tions

4

cos

8

=

Ir

+

(22)

where

tan

8

₌

₌

(1

+

_!_)

Ir

+

+

s

(23)

when the values of Equations 17 and 20 are used.

By trigonometric substitutions

Is

Im

=

J ,__

1

+

+ -

and

X

I I + X I = I cos

8

=

(I

~

~

sK Is

I r = ~[-l_+_s_2_K_2_(_1_+-~;~) 2--=]- ~

Similarly

sK Iw + Ir = I + _s_ I = Is cos

8

w X w 1

+

(1

+

~

)

(25)

(26)

The average pressure developed in the pump duct per unit of length

in the fully wound region of the induction pump is simply Ir Bm

Bm2

ilB :aJrwJ'L Ir Bm 1-Lo SVs

p

₌

=

=

=

4Tr tg 2

_p

Eliminating Im and Ir in Equation 27 using Equations 24 and 25 yields

p

=

0 ).

4., tg

sK I 2 _s

(27)

(28)

When the average pressure p (Equation 28) is maximized with respect to

s, the maximum value of p occurs when

(29)

The ideal power output per unit length in the liquid metal is

and the actual total power output in the liquid metal after accounting

for hydraulic losses or after determination of (P) _{actua 1 from}

experi-mental data is

(Po>actual

=

Thus the actual operating efficiency is

("J) actual _{total input power} (Po> actual (32)

The ideal power loss per unit length in the liquid metal, due to

r

losses of the induced currents, is

I

2

p

p

₌

=

₌

../2

p2

p

The ideal input power to the liquid metal is

(34)

Also the ideal power loss per unit length in the pump duct, due to the

r

Hence the ideal electrical

"le

=

4

2

t 2 w

P.2

=

+

-s-The combined theoretical efficiency is

"'t

=

The maximum ideal electrical efficiency of

'?em

=

(35)

(36)

(37)

occurrs when the slip is

s

=

=

X)~

Likewise the slip for maximum power output, P_{0 ,} is

1

+

The non-dimensional part of Equation 28 is

sK

=

₈

+~)

(41)

and is called the head utilization factor.

When Equation 28 is substituted into Equation 30, the

non-dimen-sional part of resulting expression is

sK (1-s) _{sK (1-s) sin2}

₈

which is referred to as the output utilization factor.

The significance of the head utilization, zh, and output utilization,

z0 , factors may not be immediately obvious. Note that both factors may

be thought of as functions of two variables (i.e. slip, s, and pump

parameter, K); however, it should be re~embered that scan vary only

over the range of values from 0 to 1.0. The wall loss parameter, X,

is constant by prior selection of the duct size, duct material, and

working fluid. By maximizing zh and z₀ with respect to s or K, the

power output in the liquid metal (Equation 30) and the pressure

developed in the duct per unit length (Equation 28) will also be

zh is maximized. Likewise if high pm-1er output is required, z0 is

maximized. The requirement of high pressure output was established

in this work.

When zh is maximized with respect to slip by substituting sh for

s in Equation 41, the result is designated by zhs· Similarly maximum

z0s is used when z0 is maximized with respect to slip by substituting s0

for s in Equation 42.

The maximum values of Equations 41 and 42 with K as the variable

occur when

and these values are

1

K

=

-_..-;:.-=-s +X

1

zhk =

2 (1

+

zok = 1-s

=

2 (1

+

.JL)

'1e

(43)

(44)

(45)

In this work,since a pump with a good pressure rise was require~ the

design could be based on strictly a maximum zhs occurring at sh from

Equation 29. However, note that basing the design on the optimum K

(i.e. Kz) offers a unique method of simultaneously maximizing zh to zhk and

Z0 to zok·· A good power output yields a better overall pump efficiency.

It should be mentioned that for a given slip, s', there exists a Kz' and

a Zhk'; likewise givens" there exists a Kz" and a zhk"· If s" is greater

than s', zhk" will be greater than zhk'· The largest zhk occurrs at s

=

equivalent to zero flow rate. Figure 1, a plot of zhk and X _s from

Equation 44, shows that the design slip is a compromise between the

flow rate and pressure r~quired. From Equation 20,--s-X is equal to

Iw

I .

r

Iw

This :::hows that the ratio ~ should be between 0 and

appro:xi-mately 1.0 if the wall loss current is to be kept below the value of

the useful liquid metal current. Also note at K ,

8

z

tan

8

=

If the slip for maximum power output (Equation 40) is evaluated at

optimum K (Equation 43) and at some preselected X, it is found that this

operating point is also one of maximum ideal electrical efficiency, i.e.

s

=

=

+

X)~

K

=

=

(39)

(47)

The equation of z0 can now be simplified by substituting for s and K from

Equations 39 and 47 respectively, to

This shows that the optimum value of the output utilization factor z0 is

equal to o~e half of the ideal electrical efficiency and that both z05

and Zok occur at the same slip. In the case of the head utilization

factor, it can be shown that zhs (occurring at sh) and zhk (occurring

at K2 ) cannot occur together; thus there is not a single optimum value

Iii

a: 0.3

f2

~

Z

_I

hk-2 I

+X-s

K - K • _...;.I __

-

z

s +X

O

0 0.1 02 0.3 0.4 0.5 0.6 0.7 OS

WALL LOSS PARAMETER

DIVIDED BY SLIP,

A:

b.

s

Ir

0.9 1.0

Figure 1. The variation of the maximum head utilization factor, zhk'

B. Windings

The proper winding design will provide the field strength required

to meet the flow rate and slip used in the stator design calculations.

A particular winding configuration is checked by calculating the total

theoretical input impedance and the stator phase current at an assumed

input .voltage and slip. If the resulting ampere turns are sufficient,

this particular winding design would be acceptable. In order to find the

theoretical input impedance and stator phase current, the equivalent

circuit of the pump must be determined. Cage and Collins (11) assumed

that the equivalent circuit of an induction motor could be modified to

compensate for the case of a liquid metal rotor in a_pump duct. They

made the following modifications: the rotor leakage reactance was assumed

to be negligibly small; the power loss in the fluid channel walls was

added in parallel with the magnetizing reactance and the primary iron

loss was neglected since saturation in the iron is lo.w. An outline of

their results is shown below.

The primary resistance, Rl, per phase is

Rl Pcu Nl (MLC)

Primary leakage

=

Ac cp

reactance, xl, is obtained

2f _qwB N 2 ₁

xl

=

---1-o-=7=----where the coil end leakage is

L s1 k 2

¢

2

(49)

(50)

and the stator slot leakage is

The magnetizing reactance,

Xm•

=

o-s

(52)

(53)

The equivalent resistance of pump duct walls referred to the primary on

a per phase basis is

w Rc

=

_p

--:::"2-t;...w_-=L-When referred to the primary, the equivalent fluid resistance, R2 , is

(w

+

tL

+

0.64

nZ

(54)

(55)

Figure 2 shows the typical single phase equivalent circuit of a polyphase

flat linear induction pump with all rotor quantities referred to the

stator.

When designing windings for a flat linear induction pump, the

grad-ing of the magnetic field must be considered. It has been shown by

Blake (12) that end effects (i.e. abrupt changes in the magnetic field

at the ends of a linear pump with nongraded windings) cause the flux to

have a pulsating component. This component in turn adds a pulsating

component of current in the liquid; the current does not increase the output power but does increase the power loss in the liquid metal by the factor

(1 +

~)

Eg

IL

jXm

Rc

I

Figure 2. Single phase equivalent circuit of polyphase flat linear induction pump

R2

-

_s

added heat loss to the liquid metal is too large even at relatively

large slips. For example,at a slip of 0.8 the power loss in the liquid

metal is nearly tripled.

These abrupt changes in the magnetic field at the ends of the pump

are overcome by many different methods of grading the winding. This

gradually increases the flux from zero at the ends to the uniform value

in the middle region. The windings may be graded over the entire length

of the pump, but this necessitates the separate design of each coil.

\•lindings may also be graded over just the end regions of the pump, thus

allowing the flux in the fully wound region to be of the ideal form.

A good example of the latter method is the double layer winding with

IV. THEORETICAL DESIGN OF PUMP

Based on the design theory of flat linear induction pumps and the

immediate performance requirements, it was decided that the pump under

design should be a five pole, one slot per pole per phase, full pitch,

twelve coil, wye connected, three phase pump with a double layer winding

on just one stator. This winding configuration automatically offers two

half wound end poles and three fully wound central poles.

Initially the designer of a linear induction pump is usually given

or must select certain operating conditions such as the working fluid,

temperature, duct material, flow rate, hydraulic losses in pump duct

acceptable, and approximate output pressure required-for the intended

application. The restrictions of hydraulic losses and pump duct

cavita-tion almost fix the overall reduction in area from the pipe to the pump

duct. And since the pipe area is known, so is the duct area. The ratio

of internal duct width and thickness is also restricted; virtually

es-tablishing these dimensions and the magnetic gap. From the duct area

dnd flow rate the fluid velocity in the pump region is determined. Now

the hydraulic losses can be checked. If these losses are satisfactory,

these dimensions may be used. It would be desirable to make the duct

thickness and the magnetic gap very small for maximum field

considera-tions; however, the resulting large hydraulic losses could not be

tolerated. Conversely, making the duct square and the resulting gap

large to reduce hydraulic losses would decrease the magnetic field in

The dimensions above and equations from design theory may now be

used to design an optimum stator structure. The optimum structure offers

maximum pressure development, maximum power, or maximum efficiency at the

given operating conditions. In this work the requirement of maximum

pressure development at a good efficiency was selected on the basis of

the intended installation of the pump in an operational sodium loop. The

factor zhs was not used since only the pressure output would be maximized.

Good efficiency would be obtained only by having a good power output.

Therefore, two methods of optimization~

0

=

=

=

z₀

=

=

a maximum zh for high pressure output and a maximum z0 for good power

output after assuming or calculating the operating slip. The calculated

slip by the first method was found to be unsatisfactory for these pump

conditions. The second method (where z₀ = z₀k,zh = zhk• and K = Kz)was

the one actually used to design the flat linear induction pump.

The design of the pump was accomplished in two steps. First, as

discussed in the two preceding paragraphs, the optimum stator structure

for sodium at 400 C was established from calculations for maximum

utiliza-tion factors. Second, the parameters for the flat linear induction pump

equivalent circuit were determined for several winding configurations.

Based on these equivalent circuits, the winding method was determined

that most nearly offered the ampere turns assumed in the first step.

A. Optimum Stator Structure

The induction pump was designed for operation with sodium metal at

area from 0.75-inch schedule 40 inconel pipe to the duct was made slightly

less than three to one in accordance with Watt (13). Larger reductions

could cause cavitation (i.e. a condition existing when inlet pressure

falls below the vapor pressure of the fluid and resulting small cavities

of vapor collapse in the pump duct). The design performance criterion

was approximately five gallons per minute flow rate of molten sodium.

The various dimensions and material properties used were

f = 60 cps

g = 0. 794 ern

k

=

~ = 1

t

=

tw

=

w = 3.81 ern

~

=

p

Pw

u

=

In order to obtain a strong field, a small air gap, g, was selected;thus

the internal thickness, t, of the pump duct was also small. The internal

width, w, was chosen so that the ratio of _!_ would be greater than 10.

The resultant f111id velocity was approximately 260 em per second at the

design flow rate using the equation

v=~

wt

At this velocity the Reynolds number was 46,250 using the equation

vd

R _e

=

(56)

(57)

and the resulting theoretical hydraulic loss was satisfactory at 0.46 psi

per foot of pump duct.

Initially an attempt was made to optimize the output utilization

factor, z0 , for maximum output at the slip of maximum efficiency and tO

maximize the head utilization factor, zh, at Kze· Using Equation 21

the wall loss parameter, X, is 0.223. The slip, found from Equation 39

with X

=

=

=

=

=

is determined from Equation 47 with X

=

X

-s;-

=

=

zh. Finally the wavelength, A , is found to be 29.6 em via Equation 18.

The velocity of the magnetic field is computed from the Equation vs

=

thus vs = 1778 em per second. At se, v

=

greatly exceeds the assumed fluid velocity of 260 em per second. In

addition this high fluid velocity, equivalent to 23.4 gpm, probably

could never be attained experimentally due to the severe hydraulic losses.

Now assume that this pump was built, but let the operation be such that

v

=

I

and 42,zh

=

=

design. However, before selecting a design consider a pump operated at

the design value of slip.

At this point it was apparent that for the design to be practical

the slip must be increased, but the extent of this increase was not

known. In this case both utilization factors were maximized at optimum

K since this method allowed different slips to be tried. It was then

necessary to find a slip which satisfied the conditions of optimum K

and a liquid velocity of 260 em per second. First note that

v

vs

=

-r:s-

and therefore

). = --::-f

-:(~~

The optimum K is obtained from Equation 43. Combining Equation 18,

the pump parameter, with the Equations 43 and 59 yields the following

1

_ _..;;; __ =

X+ s _2Tr

_gp

52 - 2

8

+

+

8

(60)

8

8

where

8=

e.

Jl-o

Equation 60, a quadratic, is easily solved for the slip s

= 0.81.

v

=

shows the value of zhk• in this case 0.392, to be very near the maximum

value since X is never zero. The corresponding value of z0k was 0.078.

The optimum K (Equation 43) is now found to be 0.976. This value of

K is shown to produce a maximum zh in Figure 3 _{by plotting values of zh}

from Equation 41 at various values of K. When this value is

substi-tuted into Equation 18, the result is ~= 21.2 em. It is obvious

now that the design and operation o~ the pump at the same slip is

beneficial. The second design method yielded a higher zh and increased

Z0 by almost a factor of two. The wavelength was shorter by the latter

method which allowed the five pole pump to be 54.8 em long rather than

75.8 em long; this was a very significant result. Thus the overall

length of the five pole pump was 54.8 em with equal slot and tooth

dimensions of 1.766 em. The nominal stator width was determined by

the combined thickness of 91 laminations of 24 gauge hot rolled high

silicon magnet core iron insulated from each other with 0.004-inch mica

sheet.

The stator structure design for the linear induction pump was still

not completely determined since the slot depth was unknown. However,

all necessary dimensions were available for the design of the pump duct.

This is shown in Figure 4.

B.

The next logical step in the design of the flat linear induction

pump was the determination of the exact winding configuration (i.e. the

required number of turns per coil and the electrical connections). It

~

~0.3

z

0

ti

~0.2

§

Q

c

Ql

L&J

:J:

.5

It<

z

h •

I

+

1t< 1

(I

t

)I

• • 0.8

X • 0.223

1.0

1.5

2.0

3.0

PUMP PARAMETER, K

Figure 3. The variation of the head utilization factor with the pump parameter

w

~-"""'"•

~

! [

---

... .. ---

=

. .

~

i

- --- - s"~ ---~··---

_-r

~

-

~

d

?

~

1

L

.~

-- ---·"

4 0 •N0-000"-'·

_z.,

I I

I

I I I I ·""'

' '

---.-1""-'C.ONS\....

.,.._· ,. '/•~ ~~~';.s;.~ .~6~ c~~~~~T''·""rv ..

~- A-.A

PUMP :a&.C..T\ON

NO"T&~; I. '-e'"""Or-T""" 0~ PIOE !l>ec,TIOW~ ...,,'-'- V.etr.£.V Pat.lO•N.-, Tl.lS.

1"-la.-.T•OM .,.. pUMP bVC."T \....n-0 L-OOP·

2 . ~0\....C•a COPPa.lt. Bu . . . ._~ T~ OuMP ~ec..,-oc.­

MI.__ ..,_&1.-'T•~• 'l=OQI ... T · ••'-\.l&lk ::::.0'-0.111..,.

Figure 4. Pump duct for flat linear induction pump

w

(0.0808-inch diameter) heavy formvar insulated copper magnet wire. Since the liquid flow can be throttled to the design rate and the ultimate test

loop can withstand pressures up to 100 psi, a design gross pressure of

60 psi at 0.8 slip and 250 volts line voltage was used. Based on the

performance of other pumps the actual pressure developed by the pump was

expected to be from 40 to 80 percent of this design figure.

The required number of ampere turns per phase was determinE!d from the equation formed by the substitution of Equation 12 for Is into

Equation 28. This result, after Equation 28 is multiplied by x to

convert p to gross pressure, is

( 4 p t g n2 ~

288

J.L

o

~

) (61)

The solution of Equation 61 with p

=

=

=

n = 5 poles, ~ = 21.2 em, _{- 4-}J-Lo 1.451 x lo-7, x

=

wound region), and zh = zhk = 0.392 yields NpH I= 3,920 ampere turns

per phase.

The various constants, dimensions, and physical properties used in

the following winding design calculations are

Ac 0.0049 in. 2

As 0.0625 in. 2

f 60 cps

g 0.311 in.

kd 1.0

kp 1.0

(MLC) 12.5 in.

q 3

sl 15

ss 15

t 0.125 in.

tw 0.0625 in.

w 1.5 in.

WB = (2. 660) (0.9) in. ws = 0.695 in.

p

Pcu

Pcu

=

c

Pw

=

The first winding configuration tried was one consisting of 120 turns

per coil with one coil in series per phase and four circuits in parallel

per phase. Using Equation 49 with cp

=

N1

=

0.127 ohms. The primary leakage reactance,

x

from Equations 50, 51, and 52 where d1 = 2.16 in. and d3

=

+

designs since a space free of conductors is required for forced air

cooling of the pump. When the area of the gap, AB

=

=

substituted into Equation 53, the magnetizing reactance,

Xm•

equivalent fluid resistance, R2 , are 3.72 ohms from Equation 54 and

2.14 ohms from Equation 55 respectively.

It should be mentioned here that the assumption of negligible rotor

reactance has been shown by Blake (12) to be good if the ratio of ~

is less than 0.1. The ratio ~ for this particular induction pump

is 0.015.

The magnitude of the total pump impedance per phase,

I ztl

=

is determined from the equivalent circuit, Figure 2, using alternating

current circuit analysis at the design slip of 0.8. Dividing the phase

voltage, E~= 144 volts, by

IZrl

I,

wind-ing configuration are 8, 080 which are more than is required. The current

per coil must also be checked since the maximum allowable current for

12 gauge wire is 20 amperes. In this example the current per coil is

16.8 amperes.

Several other winding configurations were investigated before one

was found with an ampere-turn rating approaching the design value of

3,920. The results of these investigations are summarized in Table 1.

Based on the data shown in Table 1, the 60 turns per coil winding

method was obviously the only one meeting all of the specifications.

Even though this method provided slightly more than 3,920 ampere turns

per coil, it was adjudged unnecessary to further reduce the turns per

No. of

turns/coil

120

80

60

No. of

turns/coil

120

80

60

No. of series turns/phase

120

160

240

s

_I

_z-r

_I

ohms

0.8 2.14

0.8 4.04

0.8 8. 72

cp Rl ohms

4 0.127

2 0.338

1 1.01

Ec/>

volts amperes

144 67.4

144 35.6

144 16.5

xl ohms 1.08 2.08 4.11 xm ohms 2.17 3.86 8.68

Coil current amperes

16.8

17.8

16.5

Rc Rz

ohms ohms

3. 72 2.14

6.60 3.81

14.87 8.56

a-Figure 5 shows the complete flat linear induction pump design. Note that the stator without windings, sometimes called the "core", is almost

the same size as the stator with windings; however, it has no slots.

The electrical winding connections are shown pictorially in Figure 6.

C. Theoretical Performance Characteristics

The theoretical performance characteristics of the induction pump

were computed using the equivalent circuit parameters for a winding

with 60 turns per coil. The stator phase current was determined as

before except the slip was determined from the equation

Q

s

=

as the flow rate changed. In this case the phase voltage was also

varied. Thus to each value of flow rate and line voltage there was a

corresponding unique value of stator phase current.

If the stator phase current, I, is substituted into Equation 12,

the value of the stator surface current per unit length, Is, is obtained.

The dimensions and material constants are as listed on page 26. The pump parameter, K, is 0.976. The stator surface current, Is, is then used in

the evaluation of the gross pressure developed over the fully wound region

of the pump stator via Equation 28 multiplied by x = 31.8 em (i.e. the length of the fully wound region). When the hydraulic losses, Pf' in the

pump duct are deducted from the gross pressure, xp, the resulting value

is the actual theoretical pressure developed, (p) _{ac ua} t _{1 ,} by the induction

pump. Equations 31 and 32 provide the actual power output in the

~

~

~

,.

+

+

t

'

I

:-:::

.. , ..

,!"~::~

I

I

ll

· - - · - - - -Zt.!IW 'l"

• 4"LG, ~'!'UCTUD'b~CT\.OWNCO.) -,'i, ~11.-L. T~~U TOOT~

(Toe&e.&"IVE •""-!lit .... c

I C. ~.,jOt,..$

~

'

,... .

/ ·~

----1---_/

~ ~ 1

l"a,F

I

~I!TI.. .. ..,.IN.\·

T,Cf><O~~

L _ _ _ __ _ _ _ _ _ __ _ _ _ _ _ - - - 1 - - - '

=

c~··-•T) :::t:; _.

1=•8Eit""6.~R T~ p,,-1!1,,_ e.~L T CRO~·--c• 0<-1

~'

rJ

~ ~ ~

1

_~~~

_-~

_~[~v

_{~;~~~~~:~=~--T}

-I~~

••~

U

i'li:

l! ~;~=~e:c!;:v.,,, ..

ONIE. L 6.lAII\llli.TION 00.0. ~ ....OT Ell TENC)

..

~ oen .. L T~o~tt.u aooT

(TOR-e:HW•IO-~£ .. ~ • • ·~,..-.. 3TUO'T\.IC"D 80TI.t liit.:tO~·)

.~ .... ~~~

Figure 5. Stator and core design for flat linear induction pump

~·

~

.-\._ -,.~ ,_O,t.11"-llo.T•LN 'T""o":

~ CE NOT E'•TElo.~_ec

(T '0' P eoT .... PI&'-.... :

w

c

c

a

b~

c

'A.,

2

b

b a

....

c

n n n

a

c

b

a

c

c' c" c'"

Figure 6. Winding connections for flat linear induction pump